Aspects of Reflection Groups
April 11, 12, 13
The students attending the reflection
groups course did projects on topics related to finite
Coxeter groups, root systems and their applications. Here are the
details of the mini conference and the project reports.
Specht method for constructing irreducible representations
of groups of type $A_n$ and $B_n$: Representation theory of
finite groups is an important branch of mathematics. In the context
of reflection groups one can see a rich interplay between algebra
and combinatorics. In this talk I shall focus on 2 classes of
reflection groups; the symmetric groups (type $A$) and the
octahedral groups (type $B$). I will explain, in detail, the Specht
method of constructing all irreducible representations of these
groups upto equivalence.
Classification of quivers: An integral quadratic form
(called the 'Euler form') defined on directed graphs (quivers) is
intimately connected to the representation theory of some
finite-dimensional algebras. In my talk I will describe this connection
and the classification of quivers induced by the Euler form.
Symmetries of some regular polytopes: The study of regular
polytopes has a long history in mathematics. A seminal theorem of
Coxeter says that symmetry groups of such polytopes can be realized
as reflection groups. In this talk I shall describe some these
polytopes and their symmetry groups. Time permitting, I will also
outline their classification.
Construction and symmetries of Icosahedron: The icosahedron
is one of the important platonic solids in Euclidean geometry,
admitting a highly rich group of symmetries. In this talk I shall
give an outline of Taylor's method of constructing an Icosahedron. A
brief sketch of Kepler's method will also be given and then I shall
establish that the polytope we construct is unique up to an
isometry. In the end, some nice geometrical properties of the
icosahedron and a few interesting facts about its symmetry group
will be discussed, with special emphasis on the
orientation-preserving or rotational symmetries.
An introduction to triangle groups:
A triangle group is a (possibly) inﬁnite reﬂection group. It is
realized, geometrically, by sequences of reﬂection across the sides
of a triangle. There are three types of
triangle groups; Euclidean,Spherical and Hyperbolic. In this talk I
will describe the Euclidean and Spherical triangle groups. Time
permitting, I will also talk about hyperbolic triangle groups.
The three reflections theorem: It is well known that an isometry
of an $n$-dimensional Euclidean space is a product of at most $n+1$ reflections.
Aim of my talk is to show that a similar statement is true in the context of hyperbolic
plane. I will explain the upper-half plane model and the required concepts from
hyperbolic geometry. Finally, conclude that a hyperbolic isometry is a product of at most
$3$ hyperbolic reflections.
Permutahedra and matrix mutations: For a reflection group W,
the associated W-permutahedron is the convex hull of the W-orbit of
a generic point. In this talk, I shall first describe properties of
a W-permutahedron associated to classical root systems. Later, I
shall explain the operations of the matrix mutation using concrete
Topological aspects of braid groups: Artin groups is a class
of groups naturally associated to reflection groups. In this talk, I
will first describe their construction. Then I will concentrate on a
well known class; the braid groups. These groups are associated to
symmetric groups. They are also of independent interest because of
their connection to configuration spaces. I will state this
connection and if time permits I will also skectch the proof, by
Arnold, that these are aspherical spaces.
Reflections on manifolds: Reflection on a smooth manifold is
an involutive self-diffeomorphism whose fixed poit set is a
codimension-1 submanifold. A discrete subgroup of diffeomorphisms
generated by finitely many reflections is called as a manifold
reflection group. The situation is very similar that of a reflection
group acting on a finite-dimensional vector space. In this talk I
will first describe an analouge of reflection arrangements and then
sketch the proof of the fact that these groups have a Coxeter
Classification of semisimple Lie algebras: It was
shown in the class that the finite reflection groups are classified
using the Coxeter diagrams. In this talk I will describe how
semisimple Lie algebras are also classified using such diagrams. I
will describe, first, how to associate a Weyl group to a semisimple
Lie algebra and then how to use the classification theorem done in
An introduction to Weyl groupoids: The study of root systems
and Weyl groups naturally arise in study of Lie algebras.In the last
decades,the concept of a lie algebra has been generalized in many
different directions.Recent results on pointed Hopf algebras have
led to yet another symmetry structure,the Weyl groupoid. In my talk ,
I will give an axiomatic definition of Weyl groupoids and talk about
connections with crystallographic arrangements.